<p> It is known that for a topologically ordered state the area law for the entanglement entropy shows a negative universal additive constant contribution, &ndash;&gamma;, called the topological entanglement entropy. We theoretically study the entanglement entropy of the two-dimensional Shastry-Sutherland quantum antiferromagnet using exact diagonalization on clusters of 16 and 24 spins. By utilizing the Kitaev-Preskill construction, we extract a finite topological term, &ndash;&gamma; , in the region of bond-strength parameter space corresponding to high geometrical frustration. Thus, we provide strong evidence for the existence of an exotic topologically ordered state and shed light on the nature of this model's strongly frustrated, and long controversial, intermediate phase.</p>

<p> Historically our understanding of the microscopic world has been impeded by limitations in systems that behave classically. Even today, understanding simple problems in quantum mechanics remains a difficult task both computationally and experimentally. As a means of overcoming these classical limitations, the idea of using a controllable quantum system to simulate a less controllable quantum system has been proposed. This concept is known as quantum simulation and is the origin of the ideas behind quantum computing. </p><p> In this thesis, experiments have been conducted that address the feasibility of using devices with a circuit quantum electrodynamics (cQED) architecture as a quantum simulator. In a cQED device, a superconducting qubit is capacitively coupled to a superconducting resonator resulting in coherent quantum behavior of the qubit when it interacts with photons inside the resonator. It has been shown theoretically that by forming a lattice of cQED elements, different quantum phases of photons will exist for dierent system parameters. In order to realize such a quantum simulator, the necessary experimental foundation must rst be developed. Here experimental eorts were focused on addressing two primary issues: 1) designing and fabricating low disorder lattices that are readily available to incorporate superconducting qubits, and 2) developing new measurement tools and techniques that can be used to characterize large lattices, and probe the predicted quantum phases within the lattice. </p><p> Three experiments addressing these issues were performed. In the rst experiment a Kagome lattice of transmission line resonators was designed and fabricated, and a comprehensive study on the effects of random disorder in the lattice demonstrated that disorder was dependent on the resonator geometry. Subsequently a cryogenic 3-axis scanning stage was developed and the operation of the scanning stage was demonstrated in the final two experiments. The rst scanning experiment was conducted on a 49 site Kagome lattice, where a sapphire defect was used to locally perturb each lattice site. This perturbative scanning probe microscopy provided a means to measure the distribution of photon modes throughout the entire lattice. The second scanning experiment was performed on a single transmission line resonator where a transmon qubit was fabricated on a separate substrate, mounted to the tip of the scanning stage and coupled to the resonator. Here the coupling strength of the qubit to the resonator was mapped out demonstrating strong coupling over a wide scanning range, thus indicating the potential for a scanning qubit to be used as a local quantum probe.</p>

<p> Quantum computers will potentially outperform classical computers for certain applications by employing quantum states to store and process information. However, algorithms using quantum states are prone to errors through continuous decay, posing unique challenges to engineering a quantum system with enough quantum bits and sufficient controls to solve interesting problems. A promising platform for implementing quantum computers is that of circuit quantum electrodynamics (cQED) using superconducting qubits. Here, two energy levels of a resonant circuit endowed with a Josephson junction serve as the qubit, which is coupled to a microwave-frequency electromagnetic resonator. Modern quantum circuits are reaching size and complexity that puts extreme demands on input/output connections as well as selective isolation among internal elements. Continued progress will require adapting sophisticated 3D integration and RF packaging techniques found in today's high-density classical devices to the cQED platform. This novel technology will take the form of multilayer microwave integrated quantum circuits (MMIQCs), combining the superb coherence of three-dimensional structures with the advantages of lithographic integrated circuit fabrication. Several design and fabrication techniques are essential to this new physical architecture, notably micromachining, superconducting wafer bonding, and out-of-plane qubit coupling. This thesis explores these techniques and culminates in the design, fabrication, and measurement of a two-cavity/one-qubit MMIQC featuring qubit coupling to a superconducting micromachined cavity resonator in silicon wafers. Current prototypes are extensible to larger scale MMIQCs for scalable quantum information processing.</p><p>

Superconducting qubits are a promising technology for building a scalable quantum computer. An important architecture employed in the field is called Circuit Quantum Electrodynamics (circuit QED), where such qubits are combined with high quality microwave cavities to study the interaction between artificial atoms and single microwave photons. The ultra-strong coupling achieved in these systems allows for control and readout of the quantum state of qubits to perform quantum information processing. The work on circuit QED performed in this thesis consisted of realizing an experimental setup for qubit experiments in a new laboratory, investigating the coherence and decay of higher energy levels of superconducting transmon qubits and finally demonstrating a novel coaxial form of circuit QED. Designing and building a 3D circuit QED setup involved the following main accomplishments: producing high quality 3D cavities; designing and installing the cryogenic microwave setup as well as the room temperature amplification and data acquisition circuitry; successfully developing a recipe for the fabrication of Josephson junctions; controlling and measuring superconducting 3D transmon qubits at 10mK. Several qubits were fully characterised and have shown coherence times of several microseconds and relaxation times up to 25&mu;s. Superconducting qubits in fact possess higher energy levels that can provide significant computational advantages in quantum information applications. In experiments performed at MIT, preparation and control of the five lowest states of a transmon qubit was demonstrated, followed by an investigation of the phase coherence and decay dynamics of these higher energy levels. The decay was found to proceed mainly sequentially with relaxation times in excess of 20&mu;s for all transitions. A direct measurement of the charge dispersion of these levels was performed to explore their characteristics of dephasing. This experiment was also reproduced on a 3D transmon fabricated and measured in Oxford, where due to a higher effective qubit temperature a multi-level decay model including thermal excitations was developed to explain the observed relaxation dynamics. Finally, a coaxial transmon, which we name the coaxmon, is presented and measured with a coaxial LC readout resonator and input/output coupling ports placed inline along the third dimension. This novel coaxial circuit QED architecture holds great promise for developing a scalable planar grid of qubits to build a quantum computer.